Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial

TL;DR

This paper extends graph duality on surfaces to include duality with respect to edge subsets, relating signed Bollobas-Riordan polynomials of dual graphs and unifying results connecting Jones polynomials and Bollobas-Riordan polynomials.

Contribution

It introduces a generalized duality concept for embedded graphs, linking their signed Bollobas-Riordan polynomials and unifying various link polynomial results.

Findings

Established a relation between signed Bollobas-Riordan polynomials of dual graphs.

Unified multiple recent results connecting Jones polynomial and Bollobas-Riordan polynomial.

Extended duality concepts to graphs embedded in different surfaces.

Abstract

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobas-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobas-Riordan polynomials.